Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains

Abstract

We consider the nonlinear and nonlocal problem $$ A_{1/2}u=|u|^{2^\sharp-2}u\ \text{in Ω, \quad u=0 \text{on} \partialΩ$$where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^n$, $n\ge 2$ and $2^{♯}=2n/(n-1)$ is the critical trace-Sobolev exponent. We assume that $\Omega$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{x\in {\mathbb{R}}^n\text{= s.t.}{R}_1<|x|<R_2\}\subset \Omega$ and $0\notin \Omega$, and invariant under a group $\Gamma$ of orthogonal transformations of ${\mathbb{R}}^n$ without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if $R_1/R_2$ is arbitrary and the minimal $\Gamma$-orbit of $\Omega$ is large enough, or if $R_1/R_2$ is small enough and $\Gamma$ is arbitrary.

References in corpus (1)

• Positive solutions of nonlinear problems involving the square root of the Laplacian